Chapter 7
Partially Balanced Incomplete Block Design (PBIBD)
The balanced incomplete block designs have several advantages. They are connected designs as well
as the block sizes are also equal. A restriction on using the BIBD is that they are not available for all
parameter combinations. They exist only for certain parameters. Sometimes, they require a large
number of replications also. This hampers the utility of the BIBDs. For example, if there are v 8
treatments and block size is k 3 (i.e., 3 plots in each block) then the total number of required blocks
8
are b 56 and so using the relationship bk vr , the total number of required replicates is
3
bk
r 21.
v
Another important property of the BIBD is that it is efficiency balanced. This means that all the
treatment differences are estimated with the same accuracy. The partially balanced incomplete block
designs (PBIBD) compromise on this property up to some extent and help in reducing the number of
replications. In simple words, the pairs of treatments can be arranged in different sets such that the
difference between the treatment effects of a pair, for all pairs in a set, is estimated with the same
accuracy. The partially balanced incomplete block designs remain connected like BIBD but no more
balanced. Rather they are partially balanced in the sense that some pairs of treatments have the same
efficiency whereas some other pairs of treatments have the same efficiency but different from the
efficiency of earlier pairs of treatments. This will be illustrated more clearly in the further discussion.
Before describing the set up of PBIBD, first, we need to understand the concept of “Association
Scheme”. Instead of explaining the theory related to the association schemes, we consider here some
examples and then understand the concept of an association scheme. Let there be a set of v
treatments. These treatments are denoted by the symbols 1, 2,…, v .
Partially Balanced Association Schemes
A relationship satisfying the following three conditions is called a partially balanced association
scheme with m-associate classes.
(i) Any two symbols are either first, second,…, or mth associates and the relation of
associations is symmetrical, i.e., if the treatment A is the ith associate of treatment B , then
B is also the ith associate of treatment A.
Analysis of Variance | Chapter 7 | Partial. BIBD | Shalabh, IIT Kanpur
1
, (ii) Each treatment A in the set has exactly ni treatments in the set which are the ith associate
and the number ni (i 1, 2,..., m ) does not depend on the treatment A.
(iii) If any two treatments A and B are the ith associates, then the number of treatments which
i
are both jth associate A and kth associate of B is p jk and is independent of the pair of ith
associates A and B .
The numbers v, n1, n2 ,..., nm , p jk (i, j, k 1,2,..., m) are called the parameters of m -associate partially
i
balanced scheme.
We consider now the examples based on rectangular and triangular association schemes to understand
the conditions stated in the partially balanced association scheme.
Rectangular Association Scheme
Consider an example of m 3 associate classes. Let there be six treatments denoted as 1, 2, 3, 4, 5
and 6. Suppose these treatments are arranged as follows:
1 2 3
4 5 6
Under this arrangement, with respect to each symbol, the
two other symbols in the same row are the first associates.
One another symbol in the same column is the second associate and
remaining two symbols are in the other row are the third associates.
For example, with respect to treatment 1,
treatments 2 and 3 occur in the same row, so they are the first associates of treatment 1,
treatment 4 occurs in the same column, so it is the second associate of treatment 1 and
the remaining treatments 5 and 6 are the third associates of treatment 1 as they occur in the other
(second) row.
Similarly, for treatment 5,
treatments 4 and 6 occur in the same row, so they are the first associates of treatment 5,
treatment 2 occurs in the same column, so it is the second associate of treatment 5 and
remaining treatments 1 and 3 are in the other (second) row, so they are the third associates of
treatment 5.
Analysis of Variance | Chapter 7 | Partial. BIBD | Shalabh, IIT Kanpur
2
, The table below describes the first, second and third associates of all the six treatments.
Treatment First Second Third
number associates associates associates
1 2, 3 4 5, 6
2 1, 3 5 4, 6
3 1, 2 6 4, 5
4 5, 6 1 2, 3
5 4, 6 2 1, 3
6 4, 5 3 1, 2
Further, we observe that for treatment 1, the
o number of first associates (n1 ) 2,
o number of second associates (n2 ) 1, and
o the number of third associates (n3 ) 2 .
The same values of n1 , n2 and n3 hold true for other treatments also.
Now we understand the condition (iii) of the definition of partially balanced association schemes
related to p jk .
i
Consider treatments 1 and 2. They are the first associates (which means i = 1), i.e. treatments 1 and 2
are the first associate of each other; treatment 6 is the third associate (which means j 3) of treatment
1 and also the third associate (which means k 3) of treatment 2. Thus the number of treatments
which are both, i.e., the jth (j = 3) associate of treatment A (here A 1) and kth associate of treatment
B (here B 2) are ith (i.e., i = 1) associate is pijk p33
1
1.
Similarly, consider the treatments 2 and 3 which are the first associate ( which means i 1);
treatment 4 is the third (which means j 3) associate of treatment 2 and treatment 4 is also the third
1
(which means k 3) associate of treatment 3. Thus p33 1.
Other values of p jk (i, j, k 1,2,3) can also be obtained similarly.
i
Remark: This method can be used to generate a 3-class association scheme in general for mn
treatments (symbols) by arranging them in m -row and n -columns.
Analysis of Variance | Chapter 7 | Partial. BIBD | Shalabh, IIT Kanpur
3
Partially Balanced Incomplete Block Design (PBIBD)
The balanced incomplete block designs have several advantages. They are connected designs as well
as the block sizes are also equal. A restriction on using the BIBD is that they are not available for all
parameter combinations. They exist only for certain parameters. Sometimes, they require a large
number of replications also. This hampers the utility of the BIBDs. For example, if there are v 8
treatments and block size is k 3 (i.e., 3 plots in each block) then the total number of required blocks
8
are b 56 and so using the relationship bk vr , the total number of required replicates is
3
bk
r 21.
v
Another important property of the BIBD is that it is efficiency balanced. This means that all the
treatment differences are estimated with the same accuracy. The partially balanced incomplete block
designs (PBIBD) compromise on this property up to some extent and help in reducing the number of
replications. In simple words, the pairs of treatments can be arranged in different sets such that the
difference between the treatment effects of a pair, for all pairs in a set, is estimated with the same
accuracy. The partially balanced incomplete block designs remain connected like BIBD but no more
balanced. Rather they are partially balanced in the sense that some pairs of treatments have the same
efficiency whereas some other pairs of treatments have the same efficiency but different from the
efficiency of earlier pairs of treatments. This will be illustrated more clearly in the further discussion.
Before describing the set up of PBIBD, first, we need to understand the concept of “Association
Scheme”. Instead of explaining the theory related to the association schemes, we consider here some
examples and then understand the concept of an association scheme. Let there be a set of v
treatments. These treatments are denoted by the symbols 1, 2,…, v .
Partially Balanced Association Schemes
A relationship satisfying the following three conditions is called a partially balanced association
scheme with m-associate classes.
(i) Any two symbols are either first, second,…, or mth associates and the relation of
associations is symmetrical, i.e., if the treatment A is the ith associate of treatment B , then
B is also the ith associate of treatment A.
Analysis of Variance | Chapter 7 | Partial. BIBD | Shalabh, IIT Kanpur
1
, (ii) Each treatment A in the set has exactly ni treatments in the set which are the ith associate
and the number ni (i 1, 2,..., m ) does not depend on the treatment A.
(iii) If any two treatments A and B are the ith associates, then the number of treatments which
i
are both jth associate A and kth associate of B is p jk and is independent of the pair of ith
associates A and B .
The numbers v, n1, n2 ,..., nm , p jk (i, j, k 1,2,..., m) are called the parameters of m -associate partially
i
balanced scheme.
We consider now the examples based on rectangular and triangular association schemes to understand
the conditions stated in the partially balanced association scheme.
Rectangular Association Scheme
Consider an example of m 3 associate classes. Let there be six treatments denoted as 1, 2, 3, 4, 5
and 6. Suppose these treatments are arranged as follows:
1 2 3
4 5 6
Under this arrangement, with respect to each symbol, the
two other symbols in the same row are the first associates.
One another symbol in the same column is the second associate and
remaining two symbols are in the other row are the third associates.
For example, with respect to treatment 1,
treatments 2 and 3 occur in the same row, so they are the first associates of treatment 1,
treatment 4 occurs in the same column, so it is the second associate of treatment 1 and
the remaining treatments 5 and 6 are the third associates of treatment 1 as they occur in the other
(second) row.
Similarly, for treatment 5,
treatments 4 and 6 occur in the same row, so they are the first associates of treatment 5,
treatment 2 occurs in the same column, so it is the second associate of treatment 5 and
remaining treatments 1 and 3 are in the other (second) row, so they are the third associates of
treatment 5.
Analysis of Variance | Chapter 7 | Partial. BIBD | Shalabh, IIT Kanpur
2
, The table below describes the first, second and third associates of all the six treatments.
Treatment First Second Third
number associates associates associates
1 2, 3 4 5, 6
2 1, 3 5 4, 6
3 1, 2 6 4, 5
4 5, 6 1 2, 3
5 4, 6 2 1, 3
6 4, 5 3 1, 2
Further, we observe that for treatment 1, the
o number of first associates (n1 ) 2,
o number of second associates (n2 ) 1, and
o the number of third associates (n3 ) 2 .
The same values of n1 , n2 and n3 hold true for other treatments also.
Now we understand the condition (iii) of the definition of partially balanced association schemes
related to p jk .
i
Consider treatments 1 and 2. They are the first associates (which means i = 1), i.e. treatments 1 and 2
are the first associate of each other; treatment 6 is the third associate (which means j 3) of treatment
1 and also the third associate (which means k 3) of treatment 2. Thus the number of treatments
which are both, i.e., the jth (j = 3) associate of treatment A (here A 1) and kth associate of treatment
B (here B 2) are ith (i.e., i = 1) associate is pijk p33
1
1.
Similarly, consider the treatments 2 and 3 which are the first associate ( which means i 1);
treatment 4 is the third (which means j 3) associate of treatment 2 and treatment 4 is also the third
1
(which means k 3) associate of treatment 3. Thus p33 1.
Other values of p jk (i, j, k 1,2,3) can also be obtained similarly.
i
Remark: This method can be used to generate a 3-class association scheme in general for mn
treatments (symbols) by arranging them in m -row and n -columns.
Analysis of Variance | Chapter 7 | Partial. BIBD | Shalabh, IIT Kanpur
3
